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Chaos in the Classroom: an Application of Chaos Theory DOCUMENT RESUME ED 413 289 SP 037 568 AUTHOR Trygestad, JoAnn TITLE Chaos in the Classroom: An Application of Chaos Theory. PUB DATE 1997-03-00 NOTE 17p.; Paper presented at the Annual Meeting of the American Educational Research Association (Chicago, IL, March 24-28, 1997) . PUB TYPE Information Analyses (070) Speeches/Meeting Papers (150) EDRS PRICE MF01/PC01 Plus Postage. DESCRIPTORS *Cognitive Processes; Educational Theories; Elementary Secondary Education; Higher Education; *Interaction; Learning Theories; Literature Reviews; Models; *Piagetian Theory; Teaching methods IDENTIFIERS *Chaos Theory; *Fractals; Piaget (Jean) ABSTRACT A review of studies on chaos theory suggests that some elements of the theory (systems, fractals, initial effects, and bifurcations) may be applied to classroom learning. Chaos theory considers learning holistic, constructive, and dynamic. Some researchers suggest that applying chaos theory to the classroom enhances learning by reinforcing systemic approaches to human interactions, encouraging cultural diversity as beneficial, and reaffirming theoretical notions of intelligence as dynamically multidimensional without linear progression. Other researchers believe that chaos theory cannot be applied to human learning systems; instead many of these researchers suggest social constructivism as a more appropriate model. The paper demonstrates applications of chaos theory using systems, fractals, initial effects, and bifurcations. A final section discusses models of learning, highlighting Piagetian theory and theoretical models. The paper concludes that more important than a model is the development of a perspective encompassing both the theory and its applications, and that researchers should explore the application of chaos theory to classroom learning before trying to construct a satisfactory model. (SM) ******************************************************************************** * Reproductions supplied by EDRS are the best that can be made * * from the original document. * ******************************************************************************** Chaos in the Classroom: An Application of Chaos Theory by Jo Ann Trygestad University of Minnesota U.S. DEPARTMENT OF EDUCATION PERMISSION TO REPRODUCE AND Office of Educational Research and Improvement DISSEMINATE THIS MATERIAL EDUCATIONAL RESOURCES INFORMATION HAS BEEN GRANTED BY CENTER (ERIC) 0 This document has been reproduced as received from the person or organization originating it. 0 Minor changes have been made to improve reproduction quality. Points of view or opinions stated in this docu- TO THE EDUCATIONAL RESOURCES ment do not necessarily represent official OE RI position or policy. INFORMATION CENTER (ERIC) Paper presented at a roundtable session of Chaos and Complexity Theory SIG American Educational Research Association Chicago, March 24-28, 1997 2 BEST COPY OVAI k." LE Chaos in the Classroom: An Application of Chaos Theory When a butterfly flutters in Brazil, it can cause a storm system in Texas. The "butterfly effect", was graphically recorded by meteorologist Edward Lorenz in 1961 to explain weather forecasting. The analogy of initial effects, small disturbances causing large effects, is particularly powerful. As a result, the analogy is applied to natural and human systems other than the meteorological system. Chaos theory was synthesized theoretically by Prigogine and Stengers (1984) and popularized by James Gleick (1987). As a result, the general public was introduced to systems theory and dynamic equilibrium in which a condition seeks stability through constant change. Concepts of complexity, variability, and unpredictability have replaced notions of simplicity, regularity, and predictability. Chaos, a simplification of the theoretical construct, can be defined as an event, behavior, or process which is variable, nonlinear, and unpredictable.Although chaos exists with identifiable patterns and boundaries, the patterns as well as the boundaries are flexible and indeterministic, changing unpredictably (Pool, 1989). The importance of chaos theory is its explanatory power to understand the behavior of diverse systems. What has, through observation and experimentation, seemed random and unpredictable and, therefore, been categorized as error or divergence, is now understood as representative of patterned behavior. Thus, chaos theory will be enhanced and applications will become more frequent and diverse as chaos is perceived beneficial to examine patterns within a system. The premise of this paper is the "butterfly effect", the sensitive dependence on initial conditions, and other theoretical elements of chaos theory including systems, fractals, and bifurcations, may be applied to the educational system and classroom learning.Instead of dissecting and analyzing components of learning, chaos theory suggests learning is holistic, constructive, and dynamic. Theoretical Assumptions Chaos theory can be described by its theoretical elements.Major elements of chaos are systems, fractals, initial effects, and bifurcations, which are summarized. Systems Chaos is characterized by several features of a systemit exists in nonlinear, open systems which may be simple or complex, random or stable. Chaos theory applies to nonlinear, unpredictable systems; since most systems are nonlinear and unpredictable, chaos exists in nearly all natural and human systems (Duit & Komorek, 1994).Chaos, avoided because it was thought unreliable, uncontrollable and unpredictable, is the condition of the world and must be explored. Chaos theory is also the perception of the world as an open system. The world is composed of interrelated parts; therefore, change in one area creates change in another. As with ripples in the water or movement of a crowd, change is ongoing with unpredictable results.Patterns exist and are identifiable, but they are random in both 3 their composition and connections creating unpredictable patterns.In contrast, closed systems, such as thermodynamics, are predictable and have constant energy and stability. The scientific community accumulates greater quantities of information in order to predict phenomena. For example, science can relate cause and effect to predict celestial movements of eclipses, moon phases, and sunrise and sunset; however, meteorological predictions of temperatures, precipitations, and storm conditions are limited. Although some systems are the sum of its parts, most systems are open and cannot be predicted through the accumulation of information.Since most systems are open and unpredictable, scientific thinking has gone through a revolution which affects many fields (Crutchfield, Farmer, Packard, & Shaw, 1986). Feigenbaum, a theoretical physicist, identified the universality of systems. That is, complexity has universal behavior in all systems. Feigenbaum constructed mathematical programs as proofs of the similarity of patterns in natural and human systems and so presented chaos theory to the scientific community. According to Crutchfield et al. (1986), the existence of chaos affects the scientific method itself. Systems may exhibit varying degrees of simplicity or complexity. What may appear complex is often simple; in reverse, what may appear simple is often complex. For example, chaos has been found in complex systems such as the human heart and in simple systems such as a dripping faucet.Stability was thought to exist in both systems. However, the healthy heart beats randomly and the unhealthy heart beats consistently. The dripping faucet may appear regular with ordered droplet& yet randomness occurs at the micro level and unpredictable patterns occur.Therefore, both complex and simple systems exhibit randomness. Order exists in random systems as chaos exists in stable systems. Chaotic systems appear random and fluid, yet they have underlying order and pattern (Ditto & Pecora, 1993).Natural and human systems depend on energy for sustenance; therefore, systems remain in dynamic equilibriumthe process of disequilibrium searching for equilibrium. When the system attains a nonhomogeneous, ordered state, that is the "order out of chaos" identified by Prigogine and Stengers (1984). What seems complex is simple and what seems simple is complex. What seems random is stable and what seems stable is random. What is locally unpredictable is globally stable.Since change is constant, systems are dynamic and unpredictable. Fractals Mandelbrot's mathematical principle of self-similarity was modeled using computer-generated images of a coastline to show similar patterns at any scale. The resulting theory of infinity of patternization based on scale, in which macro and micro levels replicate one another, was proposed.Mandelbrot thus created the theory of fractals. What appears as a particular pattern exists despite the scale; that is, whether the scale is large or small, the pattern continues.Mandelbrot pursued other examples of fractals including price charts and river charts, in which patterns remain constant at various scales.Microscopic examination might identify random patterns whereas the macroscopic view might see unity and cohesion; or the reverse may be true. Therefore, chaotic patterns may exhibit order or disorder in surface structure or deep structure which may be stable or oscillating (Gleick, 1987). 4 Initial Effects The "butterfly effect", the mathematical graphic of weather forecasting, shows pattern in the midst of unpredictability and, therefore, illustrates the dynamic chaos of initial states.Sensitivity
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